I would, naturally, use Forest because I happen to like it. It is very powerful and extremely flexible. I would load it with the linguistics library defaults
\usepackage[linguistics]{forest}
use a TikZ library for the wiggly line
\usetikzlibrary{decorations.pathmorphing}
and define a couple of styles for convenience.
\forestset{%
/tikz/squiggly/.style={decorate, decoration={snake, amplitude=.5mm, segment length=2mm}},
A TikZ style for wiggly lines.
wiggly/.style={edge=squiggly},
A Forest style for wiggly edges.
move me/.style n args=2{%
before drawing tree={%
x+=#1,
y+=#2,
},
},
A Forest style for moving nodes to strange positions, overriding the locations calculated by the packing mechanism.
}
no edge can be used to block an edge and phantom can be used to create invisible nodes which just help with positioning.
Then the first tree can be drawn with
\begin{forest}
[root
[un, no edge]
[dau, wiggly, no edge, move me={-5pt}{-20pt}]
[tri, wiggly, move me={-10pt}{-40pt}]
[pedwar, wiggly, move me={-15pt}{-60pt}]
[, phantom, calign with current]
[pump
[chwech, roof]
]
]
\end{forest}

Substituting the actual content
\begin{forest}
[D\textsubscript{[+DEF,+F,-PL]}
[\emph{un} <D\textsubscript{[-DEF,-F,-PL]}>, no edge, move me={200pt}{0pt}]
[\emph{une} <D\textsubscript{[-DEF,+F,-PL]}>, wiggly, no edge, move me={100pt}{-20pt}]
[\emph{le} <D\textsubscript{[+DEF,-PL]}>, wiggly, move me={60pt}{-40pt}]
[\emph{la} <D\textsubscript{[+DEF,+F,-PL]}>, wiggly, move me={15pt}{-60pt}]
[, phantom, calign with current]
[NP
[école, roof]
]
]
\end{forest}
produces

The second diagram is not a tree, but it is a tree except for one edge, so I'd just use tikz to add the extra line.
[I, tikz+={\draw [squiggly] (.parent anchor) -- (!sL.child anchor); }
To get the final nodes aligned, I would use tier and, since there's a regularity here, I'd add wiggly for all terminal nodes and use a phantom to prevent the single exception being aligned at the bottom of the tree.
Then
\begin{forest}
where n children=0{
wiggly,
tier=terminus
}{},
[A
[B
[D
[F
[H
[J
[,phantom]
[K]
]
]
[I, tikz+={\draw [squiggly] (.parent anchor) -- (!sL.child anchor); }
[,phantom]
]
]
[G
[L]
]
]
[E
[M]
]
]
[C
[N]
]
]
\end{forest}
will produce the second non-tree.

Shamelessly plugging Alan Munn's content into the second structure above,
\begin{forest}
where n children=0{
wiggly,
tier=terminus
}{},
[T
[Asp
[Voice
[$v$
[V
[$\sqrt{\text{EAT}}$
[,phantom]
[fayo]
]
]
[$v$, tikz+={\draw [squiggly] (.parent anchor) -- (!sL.child anchor); }
[,phantom]
]
]
[Voice\\\textminus act
[o]
]
]
[Aspect\\+perf
[ik]
]
]
[{T\\+past,1sg}
[a]
]
]
\end{forest}
produces

Complete code:
\documentclass[border=10pt,multi,tikz]{standalone}
\usepackage[linguistics]{forest}
\usetikzlibrary{decorations.pathmorphing}
\usepackage{amsmath,textcomp}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\begin{document}
\forestset{%
/tikz/squiggly/.style={decorate, decoration={snake, amplitude=.5mm, segment length=2mm}},
wiggly/.style={edge=squiggly},
move me/.style n args=2{%
before drawing tree={%
x+=#1,
y+=#2,
},
},
}
\begin{forest}
[root
[un, no edge]
[dau, wiggly, no edge, move me={-5pt}{-20pt}]
[tri, wiggly, move me={-10pt}{-40pt}]
[pedwar, wiggly, move me={-15pt}{-60pt}]
[, phantom, calign with current]
[pump
[chwech, roof]
]
]
\end{forest}
\begin{forest}
[D\textsubscript{[+DEF,+F,-PL]}
[\emph{un} <D\textsubscript{[-DEF,-F,-PL]}>, no edge, move me={200pt}{0pt}]
[\emph{une} <D\textsubscript{[-DEF,+F,-PL]}>, wiggly, no edge, move me={100pt}{-20pt}]
[\emph{le} <D\textsubscript{[+DEF,-PL]}>, wiggly, move me={60pt}{-40pt}]
[\emph{la} <D\textsubscript{[+DEF,+F,-PL]}>, wiggly, move me={15pt}{-60pt}]
[, phantom, calign with current]
[NP
[école, roof]
]
]
\end{forest}
\begin{forest}
where n children=0{
wiggly,
tier=terminus
}{},
[A
[B
[D
[F
[H
[J
[,phantom]
[K]
]
]
[I, tikz+={\draw [squiggly] (.parent anchor) -- (!sL.child anchor); }
[,phantom]
]
]
[G
[L]
]
]
[E
[M]
]
]
[C
[N]
]
]
\end{forest}
\begin{forest}
where n children=0{
wiggly,
tier=terminus
}{align=center},
[T
[Asp
[Voice
[$v$
[V
[$\sqrt{\text{EAT}}$
[,phantom]
[fayo]
]
]
[$v$, tikz+={\draw [squiggly] (.parent anchor) -- (!sL.child anchor); }
[,phantom]
]
]
[Voice\\\textminus act
[o]
]
]
[Aspect\\+perf
[ik]
]
]
[{T\\+past,1sg}
[a]
]
]
\end{forest}
\end{document}
pstricksapproaches. If you really need generic graphs, you need a different approach, obviously. If you just need to add the odd edge here and there, you can draw it after the main tree using TikZ.tikz-qtree,forest, TikZ's built-in support for trees or thetreeslibrary. Or the graph-drawing stuff. Or .... – cfr Aug 24 '16 at 20:29